Multiple choice

There are $2$ groups of subjects one of which consists of $5$ science subjects & $3$ engg. subjects & other consists of $3$ science & $5$ engg. subjects. An unbiased die is cast. If the number $3$ or $5$ turns up a subject is selected at random from first group, otherwise the subject is selected from $2^{nd}$ group. Find the probability that an engg. subject is selected.

  1. $\dfrac{13}{24}$
  2. $\dfrac18$
  3. $\dfrac{5}{12}$
  4. $\dfrac{11}{24}$
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A Correct answer
Explanation

P(Group 1) = P(3 or 5) = 2/6 = 1/3. P(Group 2) = 1 - 1/3 = 2/3. P(Eng | G1) = 3/8. P(Eng | G2) = 5/8. P(Eng) = (1/3 * 3/8) + (2/3 * 5/8) = 3/24 + 10/24 = 13/24.

AI explanation

Using the law of total probability, the overall probability is the sum of selecting from each group and then picking an engineering subject. The probability of selecting the first group is 2/6, and selecting engineering from it is 3/8, while the probability of selecting the second group is 4/6, and selecting engineering from it is 5/8. Calculating (2/6)(3/8) + (4/6)(5/8) yields 6/48 + 20/48, which equals 26/48 or 13/24.

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